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# Implementation and security analysis of practical quantum secure direct communication

Light: Science & Applicationsvolume 8, Article number: 22 (2019) | Download Citation

## Abstract

Rapid development of supercomputers and the prospect of quantum computers are posing increasingly serious threats to the security of communication. Using the principles of quantum mechanics, quantum communication offers provable security of communication and is a promising solution to counter such threats. Quantum secure direct communication (QSDC) is one important branch of quantum communication. In contrast to other branches of quantum communication, it transmits secret information directly. Recently, remarkable progress has been made in proof-of-principle experimental demonstrations of QSDC. However, it remains a technical feat to bring QSDC into a practical application. Here, we report the implementation of a practical quantum secure communication system. The security is analyzed in the Wyner wiretap channel theory. The system uses a coding scheme of concatenation of low-density parity-check (LDPC) codes and works in a regime with a realistic environment of high noise and high loss. The present system operates with a repetition rate of 1 MHz at a distance of 1.5 kilometers. The secure communication rate is 50 bps, sufficient to effectively send text messages and reasonably sized files of images and sounds.

## Introduction

Economic, political, and social well-being in the world depend crucially on secure communication infrastructures. Present communication is secured through encryption techniques, relying on pre-shared key and cryptographic protocols built on the computational difficulty of certain mathematical problems, for example, the RSA public key scheme1. There are potential dangers with the present secure communication system. On one hand, these cryptographic protocols are based on mathematically difficult problems that are not rigorously proven to have no efficient solution algorithms. These protocols may be broken one day, or might have been broken privately already by some genius; we do not yet know whether efficient algorithms for solving these problems exist. On the other hand, some cryptography may become insecure with the rapid development of supercomputers and the prospect of practical quantum computers2. In contrast to cryptographic algorithms, physical-layer security is based on the conditions that the eavesdropper has unlimited computing power, but the legitimate receiver has a physical advantage over the eavesdropper. In 1975, Wyner presented a degraded wiretap channel model3, which is a basic channel model when security is concerned. Secrecy capacity is defined as the supremum of all the achievable transmission rates with security and reliability. For classical communication, estimation of the secrecy capacity in a practical communication system is hard, because it is difficult for the legitimate parties to detect eavesdropping. When quantum systems such as single photons or entangled pairs of photons are used to transmit digital information, quantum physics principles give rise to novel capability unachievable with classical transmission media4. It is impossible in principle for Eve to eavesdrop without disturbing the transmission so as to avoid detection. The first quantum communication protocol, proposed by Bennett and Brassard (BB84)5, showed how to exploit quantum resources for secure key agreement. Quantum-key distribution5,6,7,8,9 distributes a random key, rather than the information itself, and the information is sent through another classical communication channel.

In 2000, quantum secure direct communication (QSDC) was proposed10. QSDC can communicate information directly without key distribution10,11,12,13,14, which eliminates further security loopholes associated with key storage and ciphertext attacks15,16, offering a new tool for selection in the zoo of secure communication protocols. Recently, experiments were completed of proof-of-principle demonstrations of QSDC based on single photons17 and entangled pairs18,19. In particular, Zhang et al.19 demonstrated QSDC in a fiber over a meaningful distance of 500 m using the two-step QSDC protocols10,11.

Here, we report an experimental implementation of a practical quantum secure communication system using a protocol based on the DL04 protocol12. To move QSDC forward into practical application, a number of key issues must be solved. Security analysis of information transmission is crucial for practical application. According to Wyner’s wiretap model, it is essential to let the system work at a capacity below the secrecy capacity of the channel. We estimated the secrecy capacity using the error rate from the sampling-checking process of the system. Once this secrecy capacity estimation is completed, it is possible to design a coding scheme with a communication rate smaller than this secrecy capacity. We have developed a coding scheme using concatenation of low-density parity check (LDPC) codes20,21. The scheme is specifically designed for operating in the high loss and high error-rate regime, unique for quantum communication. The experiment shows that our QSDC platform can work effectively in a realistic environment. In our system, the single-photon source was an attenuated faint laser pulse with a repetition rate of 1 MHz. The distance was 1.5 km, and the secure information transmission rate achieved was 50 bps, sufficient to transmit text messages and image or sound files of reasonable size.

## Results

### Practical DL04-QSDC (PDL04 QSDC) protocol

Our practical quantum secure direct communication scheme is based on the DL04 protocol using single photons12. The scheme is illustrated in detail in Fig. 1. The “main channel” and the “wiretap channel” are discrete memoryless channels; the main channel represents the channel between the sender and receiver, while the wiretap channel represents the channel between the legitimate users and the eavesdropper. The protocol contains the following four steps.

1. (1)

Bob, a legitimate information receiver, prepares a sequence of qubits. Each qubit is randomly in one of the four states $$\left| 0 \right\rangle$$, $$\left| 1 \right\rangle$$, $$\left| + \right\rangle$$, and $$\left| - \right\rangle$$, where $$\left| 0 \right\rangle$$, $$\left| 1 \right\rangle$$ are the eigenstates of Pauli operator Z, and $$\left| + \right\rangle$$, $$\left| - \right\rangle$$ are the eigenstates of Pauli operator X. Then, he sends the sequence of states to the information sender Alice.

2. (2)

After receiving the single photon sequence, Alice randomly chooses some of them and measures them randomly in the Z-basis or the X-basis. She publishes the positions, the measuring basis and measurement results of those single photons. Bob compares this information with his preparations of these states, estimates the bit-error rate of the Bob-to-Alice channel, and informs Alice through a broadcast channel. Thus, Alice can estimate the maximum secrecy capacity Cs of the Bob-to-Alice channel using the wiretap channel theory.

3. (3)

Alice chooses a coding scheme for the remaining qubits. This coding scheme is based on the concatenation of LDPC codes that will be described in the discussion section. The following two unitary operations,

$$I = \left| 0 \right\rangle \left\langle 0 \right| + \left| 1 \right\rangle \left\langle 1 \right|,Y = \left| 1 \right\rangle \left\langle 0 \right| - \left| 0 \right\rangle \left\langle 1 \right|$$

map ‘0’ and ‘1’, respectively; they are further used for constructing the code words. Then, she sends them back to Bob.

4. (4)

Bob decodes Alice’s message from his received signals after measuring the qubits in the same basis he prepared them. If the error rate is below the correcting capability of the LDPC code, the transmission is successful. Then, they start again from step (1) to send another part of the secret message until they complete the transmission of the whole message. If the error rate is larger than the correcting capability of the LDPC code, neither Bob nor Eve can obtain information. In this case, they terminate the process.

### Security analysis

According to Wyner’s wiretap channel theory3, the secrecy capacity is

$$C_s = \mathop {{\max }}\limits_{\{ p\} } \left\{ {I(A:B) - I(A:E)} \right\}$$
(1)

where p represents the probability of unitary operation I. I(A:B) and I(A:E) are the mutual information between Alice and Bob and between Alice and Eve, respectively. Moreover, I(A:E) represents the maximum information that an eavesdropper can obtain using the best strategy she can.

The state Bob prepared is a complete mixed state, $$\rho = \left( {\left| 0 \right\rangle \left\langle 0 \right| + \left| 1 \right\rangle \left\langle 1 \right|} \right)/2$$, because he prepares it with equal probabilities of the four states, $$\left| 0 \right\rangle$$, $$\left| 1 \right\rangle$$, $$\left| + \right\rangle$$, $$\left| - \right\rangle$$. We consider the case of collective attack, where the most general quantum operation that Eve may perform in the forward Bob-to-Alice channel consists of a joint operation on the qubit and some ancilla that belong to Eve,

$$\rho ^{BE} = U\left( {\rho \otimes \left| \varepsilon \right\rangle \left\langle \varepsilon \right|} \right)U^ +$$
(2)

where $$\left| \varepsilon \right\rangle$$ represents Eve’s ancillary state and U is a unitary operation acting on the joint space of the ancilla and the qubit. Then, Eve resends the qubit to Alice and stores her ancilla until the qubit is sent back. Alice performs an operationIwith probability p or Y with probability 1−p. After operating by Alice, the state becomes

$$\rho ^{ABE} = p \cdot \rho _0^{BE} + \left( {1 - p} \right) \cdot \rho _1^{BE}$$
(3)

where $$\rho _0^{BE} = I\rho ^{BE}I$$ and $$\rho _1^{BE} = Y\rho ^{BE}Y^ +$$. To gain Alice’s information, Eve must distinguish Alice’s encoded qubit $$\rho _0^{BE}$$ from $$\rho _1^{BE}$$ by performing coherent measurements on any number of qubits and ancilla. The maximum mutual information between Alice and Eve is upper-bounded by:

$$I(A:E) \le \chi = \, \mathop {{\max }}\limits_{\{ U\} } \left\{ S(\rho ^{ABE}) - p \cdot S(\rho _0^{BE})\right. \\ - \left.(1 - p) \cdot S(\rho _1^{BE}) \right\}$$
(4)

where S(ρ) is the von Neumann entropy, and χ is the Holevo bound22. We obtain the maximum mutual information between Alice and Eve (the detailed derivation is given in supplementary information),

$$I(A:E) \le h(\xi )$$
(5)

where $${\xi = ( {1 - \sqrt {( {1 - 2p} )^2 + ( {1 - 2e_x - 2e_z} )^2[ {1 - ( {1 - 2p} )^2} ]} } )/2}$$, ex and ez are the bit-error rates in the X-basis and the Z-basis in the error-check, respectively, and h(x) = −x log2 x−(1–x) log2 (1–x) is the binary Shannon entropy.

Because of imperfect efficiency of the detectors and channel loss, Bob cannot receive all the qubits. Gottesman has proven the security of the Bennet-Brassard quantum-key-distribution protocol in the case in which the source and detector are under the limited control of an adversary23. Similarly, considering the detectors and channel loss, the maximum mutual information between Alice and Eve becomes

$$I(A:E) \le Q^{{\rm Eve}} \cdot h(\xi )$$
(6)

where QEve is the maximum rate at which Eve can access the qubits. Highly attenuated lasers are used as an approximate single-photon source in our implementation; for a better treatment of such an approximate single photon source, one can use the decoy state methods24,25,26.

The main channel can be modeled as a cascaded channel, which consists of a binary symmetric channel and a binary erasure channel in series27. The mutual information between Alice and Bob is,

$$I(A:B) = Q^{{\rm Bob}} \cdot \left[ {h\left( {p + e - 2pe} \right) - h(e)} \right]$$
(7)

where QBob is the receipt rate at Bob’s side and e is the bit-error rate between Alice and Bob. We can estimate the lower bound of the secrecy capacity,

$$C_{s} = \mathop {{\max }}\limits_{\{ p\} } \left\{ {I(A:B) - I(A:E)} \right\}\\ = \mathop {{\max }}\limits_{\{ p\} } \left\{ {Q^{{\rm Bob}} \cdot \left[ {h\left( {p + e - 2pe} \right) - h(e)} \right] - Q^{{\rm Eve}} \cdot h(\xi )} \right\}\\ = Q^{{\rm Bob}} \cdot \left[ {1 - h(e)} \right] - Q^{{\rm Eve}} \cdot h\left( {e_x + e_z} \right)\\ = Q^{{\rm Bob}} \cdot \left[ {1 - h(e) - g \cdot h\left( {e_x + e_z} \right)} \right]$$
(8)

where g represents the gap between QEve and QBob, depending on the back-channel loss and the efficiency of the detector.

For any wiretap channel, if the secrecy capacity is non-zero, i.e., if the legitimate receiver has a better channel than the eavesdropper, there exists some coding scheme that achieves perfect secrecy3. Not all coding schemes can guarantee the security; the security depends on the details of the coding.

### Experimental results

We implemented the above scheme in a fiber system with phase coding28. The details of the experimental setup and methods are shown in the material and methods section, and the coding scheme is described in the discussion section. In our experiment, we initially set the distance at 1.5 km, which is a typical distance between buildings in a secure area. Figure 2 shows the error rates at Alice’s and Bob’s sites; the horizontal axis is labeled with the number of blocks processed. ex and ez are the error rates of measurements using the X-basis and Z-basis at Alice’s site, respectively. We estimate the error rate block by block. Each block contains 1312 × 830 = 1,088,960 pulses, including a frame head for synchronization. Under normal working conditions, their values are ~0.8%. At Bob’s site, of the pulses he sent to Alice previously, he receives 0.3% of them; namely for every 1000 pulses, 3 photons are counted when Bob measures the returned pulses. The error rate at Bob’s site is lower than that at Alice’s site due to the intrinsic robustness of the retrace-structure of light, usually ~0.6%. Here, the mean photon number is 0.1. The inherent loss of the quantum channel is 14.5 dB, including the efficiency of the superconducting nanowire single-photon detectors, ~70%, and the optical elements, ~13 dB. Because the mean photon number is 0.1 and the channel loss of 1.5 km fiber is 0.6 dB, the total loss of the system is 25.1 dB. Shown in Fig. 3, the mutual information I(A:B) and I(A:E) versus the loss of the system are two straight lines. The area between these two lines is the information-theoretic secure area; i.e., for a coding scheme with an information rate within these areas, it is possible to guarantee the security reliably. In our experiment, the error rates are initially set at values as above, namely e is 0.6% and ex and ez are 0.8%. Then, the secrecy capacity is estimated as 0.00184 for loss at 25.1 dB. For the number N in the pseudo-random sequence, we set N = 830, after optimization. Together with the chosen error correcting code, our coding scheme gives a transmission rate 0.00096 when the bit error rate is chosen as 10−6. Additionally, $$I(A:E) = g \cdot Q^{{\rm Bob}} \cdot h\left( {e_x + e_z} \right) = 9.1 \times 10^{ - 4}$$, where the loss of the back channel, including the efficiency of the detector and channel loss, is ~4.1 dB, so that g = 2.57. This yields a secure information rate of 50 bps, which is well within the secure area in Fig. 3.

## Discussion

It is well-known that in quantum communication, photon loss is very high due to inefficient photon sources, high channel loss and low detector efficiency. To guarantee the reliability and security of transmission for QSDC, we designed a coding scheme based on the concatenation of LDPC codes, with preprocessing based on the universal hashing families (UHF)29.

Details of our coding scheme are illustrated in Fig. 4. For each message block m of length Nm, the sender, namely Alice, generates a local sequence of random bits, denoted r, of length Nr. Then, she maps (m, r) to a vector u of length Nu= Nr+ Nm, by the inverse of an appropriately chosen UHF, determined by a public random seed s. Information theoretic security can be guaranteed if the ratio of the length of the random bits to the length of the code word is higher than the mutual information between Alice and Eve30. In information theory, the noisy-channel coding theorem establishes reliable communication for any given degree of noise contamination of a communication channel31. To ensure the reliability of the information, Alice encodes the vector u to v of length Nv using the generator matrix of a specified LDPC code. Then, she maps each coded bit to a sequence of length N to obtain a transmitted sequence, namely a code word of length Nc that is transmitted over the quantum channel. According to the noisy-channel coding theorem31, the ratio of the length of the vector u to the length of the code word cannot be higher than the channel capacity. We deduce that the information rate,

$$R = \frac{{N_m}}{{N_c}} = \frac{{N_u}}{{N_c}} - \frac{{N_r}}{{N_c}} \le I(A:B) - I(A:E) \le C_s$$
(9)

After receiving the modulated pulses from Alice, the legitimate receiver Bob makes measurements in the same basis as he prepared them. Though only a fraction of photons in a pseudo-random sequence can reach Bob’s site, he can still readout the coded bit by looking at the log-likelihood ratios of each coded bit calculated from the received sequence, and he decodes the LDPC code with an iterative propagation-decoding algorithm with the log-likelihood ratios. Then, Alice announces the public random seed s, so that Bob can obtain the secure message by the certain UHF with the seed.

For our system, we consider a (1408, 1024) quasi-cyclic (QC)-LDPC code of block length Nv = 1408, which is a standardized LDPC code of the Consultative Committee for Space Data Systems (CCSDS) for use in near-earth and deep-space applications32. The last 128 coded bits in the obtained code word of this LDPC code are punctured to achieve better error-correction performance. Thus, the actual block length of punctured LDPC code word is reduced to 1280 and the actual code rate is 0.8. Then, each coded bit in the punctured LDPC code word is mapped into a pseudo-random sequence of length 830 to obtain a transmitted sequence of length Nc = 1280 × 830 = 1,062,400 such that our coding scheme has a transmission rate of 0.00096. During decoding, the log-likelihood ratio of each coded bit of LDPC code is first calculated based on its corresponding pseudo-random sequence. Then, an effective iterative propagation-decoding algorithm, the scaling Min-Sum decoding algorithm33, is used to decode this LDPC code. The maximum number of iterations and scaling factor of the scaling Min-Sum decoding algorithm are set to 65 and 0.75, respectively. This shows that the decoding bit-error rate is ~10−6 in our code scheme.

## Materials and methods

The experimental setup is shown in Fig. 5. Bob prepares a sequence of single-photon pulses. After polarization control and attenuation, the pulses go to the Mach-Zehnder ring in which a random phase of 0, π/2, π, and 3π/2, is encoded, which is equivalent to preparing qubits randomly in the $$\left| 0 \right\rangle$$, $$\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)/\sqrt 2$$, $$\left| 1 \right\rangle$$ and $$\left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)/\sqrt 2$$ states, respectively. Then, it is sent to Alice’s site through a 1.5 km-long fiber. After arriving at Alice’s site, it is separated into two parts, one goes to the encoding module, and the other goes to the control module. In the control module, the qubits are measured, and the results are compared with Bob’s through the classical communication line connecting the two FPGAs shown at the bottom of Fig. 5. Simultaneously, encoding is performed in the encoding module. If the error rate is smaller than the threshold, the encoding part is allowed to send the single photons back to Bob through the same fiber; they then are guided to the single-photon detectors, where they are measured. The three phase modulators, the single photon detectors, and the encoding of messages are controlled at the two sites by the FPGAs, which are further controlled by upper-position computers.

The advantage of such forward-backward routing of the photon pulses is the automatic compensation of the drift of the polarizations of the time-bin pulses, because they exchange their routes after reflection by the Faraday rotator at Alice’s site. This automatic compensation design was proposed by Martilelli34 and has also been used in the plug-play QKD system35. The difference between the plug-play QKD scheme and DL04-based schemes, such as in refs. 7,12,17 and in this PDL04-QSDC scheme, is in the strength of light pulses in the forward channel. In refs. 7,12,17, single photons are used in both the forward and backward channels, whereas in plug-play QKD35, the forward channel uses strong classical light pulses; only the Alice-to-Bob backward channel uses single-photon pulses. This mechanism of automatic compensation of polarization fluctuation works both at the single photon level and at the strong-intensity level; hence, it greatly enhances the interference in our scheme and leads to high visibility36. However, in the check-module of our system, such a retrace-light circuit is not applicable, and active polarization compensation must be used; namely, one monitors the drift constantly and when it reaches some value, forcibly restores them. As a result, the error rate in the check mode is usually higher than that in the communication mode.

In summary, we have implemented a practical quantum secure direct-communication system in a realistic environment of high noise and high loss. To combat error and loss, LDPC code and pseudo-random sequence techniques are applied. The security of the system is analyzed in detail using the wiretap channel theory. Given the error rates, the secrecy capacity of the channel can be estimated. When the secrecy capacity is non-zero, a coding scheme with an information rate less than the secrecy capacity will ensure both the security of the information transmission and reliability of the information. At a practical meaningful distance of 1.5 km, a secure information rate of 50 bps is achieved. These parameters are premature, and there is much room for improvement. With current technology, an information rate of a dozens of kbps is achievable.

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## Acknowledgements

This work was supported by the National Basic Research Program of China under Grant Nos. 2017YFA0303700 and 2015CB921001 and the National Natural Science Foundation of China under Grant Nos. 61727801, 11474181, 61871257, and 11774197. This work is supported in part by the Beijing Advanced Innovation Center for Future Chip (ICFC). Helpful discussions with Prof. Zhenqiang Yin and Prof. Shuang Wang are gratefully acknowledged.

### Author contributions

R.Q., Z.L., P.N., J.G. and G.-L.L. designed the protocol and the optical circuits, and setup the physical layout. Z.S., W.H., L.S., Q.H., L.Y. made the LDPC coding and pseudo-M series. R.Q., Z.L., L.Y. and G.-L.L. completed the security analysis. L.Y. and G.-L.L. supervised the project. G.-L.L. led the entire project. All authors contributed to the writing of the paper.

## Author information

### Author notes

1. These authors contributed equally: Ruoyang Qi, Zhen Sun, Zaisheng Lin

### Affiliations

1. #### State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing, 100084, China

• Ruoyang Qi
• , Penghao Niu
• , Jiancun Gao
•  & Gui-Lu Long
2. #### School of Information and Technology, Tsinghua University, Beijing, 100084, China

• Zhen Sun
• , Zaisheng Lin
• , Wentao Hao
•  & Liuguo Yin
3. #### Beijing National Research Center for Information Science and Technology, Beijing, 100084, China

• Zaisheng Lin
• , Liuguo Yin
•  & Gui-Lu Long
4. #### School of Electronic and Information Engineering, Beihang University, Beijing, 100191, China

• Liyuan Song
•  & Qin Huang
5. #### Innovative Center of Quantum Matter, Beijing, 100084, China

• Gui-Lu Long
6. #### Beijing Academy of Quantum Information Science, Beijing, 100193, China

• Gui-Lu Long

### Conflict of interest

The authors declare that they have no conflict of interest.

### Corresponding authors

Correspondence to Liuguo Yin or Gui-Lu Long.

## About this article

### DOI

https://doi.org/10.1038/s41377-019-0132-3


text:
\Delta self-energy at finite temperature and density and the \pi N cross-section

# Δ self-energy at finite temperature and density and the πN cross-section

Snigdha Ghosh Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata - 700064, INDIA Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400085, INDIA    Sourav Sarkar Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata - 700064, INDIA Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400085, INDIA    Sukanya Mitra Indian Institute of Technology Gandhinagar, Gandhinagar-382355, Gujarat, INDIA
###### Abstract

The self energy of -baryon is evaluated at finite temperature and density using the real time formalism of thermal field theory. The Dyson-Schwinger equation is used to get the exact thermal propagator followed by the spectral function of . The scattering cross section obtained using explicit exchange is normalized to the experimental data in vacuum and its medium modification is implemented by means of the exact thermal propagator. A significant suppression of the peak of the cross-section is observed at higher temperature and baryon density. Effects on the mean relaxation time of nucleons and the temperature dependence of the shear viscosity of a pion nucleon gas are demonstrated.

## I Introduction

The properties of hadrons under conditions of high temperature and/or baryon density have been widely studied owing to the possibility of restoration of chiral symmetry of QCD spontaneously broken by the vacuum Bazavov (). Relativistic collisions of heavy ions provide the unique opportunity to actually produce hot/dense matter under controlled conditions and provide experimental verification of such studies (see e.g. Rapp ()). The spectral changes of hadrons as a result of compression and/or heating thus play a very significant role in unraveling the dynamics of the underlying colour degrees of freedom and the vacuum structure of QCD CBMBook (). The broadening of vector spectral function in the medium observed through the invariant mass spectra of dileptons in heavy ion collisions at SPS Arnaldi () and RHIC Adare () have been interpreted (see e.g. Sarkar ()) to have non-trivial, though indirect implications on chiral symmetry restoration. In this connection the study of spectral properties of baryon resonances such as and etc. are important; though not measurable directly because of final state interactions, they provide valuable input to the evaluation of the self-energies of low mass vector mesons which can be measured in heavy ion collision experiments through electromagnetic probes. In addition to broad resonances stable hadrons such as the nucleon also develop considerable widths in the medium Ghosh2 () and this is mainly due to resonant scattering with pions involving baryon resonances.

The in-medium self-energy of the has played a key role in the understanding of dynamics of nuclear interactions particularly in the resonance region where it is excited as a resonance Ericson (). The propagation of , and thus become intimately connected. The description of pion-nucleus interaction for example, depends on the formation, propagation and decay of the in the nuclear environment Oset (); Malfliet (). Models based on this picture, known as -hole models, successfully explain many scattering phenomena in elastic as well as inelastic channels. The self-energy extracted using this approach has also found applications in transport models describing the dynamical evolution of nucleus-nucleus collisions Mosel (); Wolf ().

The spectral modification of the at finite temperature and density and the in-medium cross-section remains an interesting issue of discussion. A very useful and topical application of this study is to investigate the medium dependence of the shear viscosity of hot and dense hadronic matter which is produced towards the later stages of relativistic heavy ion collisions. In the kinetic theory approach of evaluating transport coefficients like shear and bulk viscosities, thermal conductivity etc. the cross-section enters as the dynamical input. Since pions account for most of the multiplicity in relativistic heavy ion collisions, the case of the pion gas has received some attention in recent times. Scattering amplitudes evaluated using chiral perturbation theory to lowest order have been used in Santalla (); Chen () and unitarization improved estimates were employed in Dobado () to evaluate the shear viscosity. Again, phenomenological scattering cross-section using experimental phase shifts were utilized in Prakash (); Chen (); Davesne (); Itakura () to obtain the viscous coefficients. While in Moore (); Juan () the effect of number changing processes on the bulk viscosity of a pion gas has been studied, in Fraile () unitarized chiral perturbation theory was used to demonstrate the breaking of conformal symmetry by the pion mass. It is important to point out that in all these approaches vacuum amplitudes have been used. However, cross-sections of scattering between excitations in the medium could be non-trivially affected by the presence of other particles constituting it. It follows that in addition to the usual quantum fluctuations, thermal fluctuations modify the propagation of mediating particles and consequently the invariant amplitude. The case of the cross-section in a hot pion gas was studied by Barz et al Barz () obtaining a substantial reduction of the magnitude in the region of the peak. With this motivating feature, recently Mitra1 (), the amplitudes for elastic scattering were evaluated using in-medium propagators for the exchanged and mesons incorporating loop graphs with , , and mesons in the internal lines Ghosh1 (). The cross-section so obtained showed a substantial reduction in magnitude at the peak position at higher values of temperature. When used in the collision integral of the transport equation, these medium modified scattering amplitudes were found to significantly affect the temperature dependence of the viscosities Mitra2 () and thermal conductivity Mitra3 () of a hot pion gas.

Based on our experience with the pion gas and keeping in view the upcoming CBM experiment at FAIR it is natural to ask how the presence of a finite baryon density in addition to temperature is likely to affect the shear viscosity. To study such effects one has to include nucleons and consequently the cross-section becomes the principal dynamical factor in its evaluation. Incorporating the in-medium cross-section calculated using the modified self-energy a more reliable estimate of the shear viscosity, in particular it’s dependence on temperature and baryon density can be obtained. When used as input in the viscous hydrodynamic equations a more realistic scenario of space time evolution of the later stages of heavy ion collisions is likely to be achieved.

In this work we obtain the self-energy at finite temperature and baryon density evaluating several one-loop diagrams with , , and in the internal lines using standard thermal field theoretic methods. The in-medium propagator of the is then used in the scattering amplitudes to obtain the cross-section. This is utilized to evaluate the relaxation times in a hadronic gas mixture of pions and nucleons. Finally, the temperature and density dependence of the shear viscosity is obtained.

Medium modifications of the resonance has been studied mostly in nuclear matter Oset (); Wehrberger (); Korpa1 (); Kim (); Xia (). Whereas a many body expansion in terms of particle-hole excitations has been used in Oset () to evaluate the self-energy in nuclear matter, in Wehrberger () and Kim () its decay in the medium is investigated using the quantum hadrodynamics (QHD) model. A self-consistent treatment of pions and ’s in nuclear matter at zero and finite temperature may be found in Xia () and Korpa2 () respectively. In Korpa1 (), the self-energy due to loop in nuclear matter is obtained in a relativistic approach. More recently, modification of the spectral function at finite temperature and density due to resonant scattering off thermal pions has been obtained in vanHees (). In addition to these theoretical studies, properties of the have been studied experimentally using invariant mass analyses of pairs Hjort (); Mishra ().

In the next section we recall some basic features of the real-time version of thermal field theory. Then we evaluate the self-energy and discuss numerical results. In the subsequent section we evaluate the amplitudes of scattering leading to the cross-section in the medium. This is followed by a section on the shear viscosity of a gas and finally by summary and discussions. Some details of the calculation is provided in the Appendix.

## Ii The Δ self-energy in the medium

### ii.1 The in-medium propagators in the real time formalism

In the real time formalism of thermal field theory, all two-point functions including the self-energy take the form of matrices Mallik (); Bellac (). But each of the matrices may be diagonalized, when it is given essentially by a single analytic function which determines completely the dynamics of the corresponding two-point function. This function being given by any one, say the 11-component of the matrix, we need to evaluate only this component of the self-energy matrix. In the following we specify only the 11-component of the thermal propagator for the particles involved in the one-loop graphs that are considered in this work.

The 11-component of a free thermal propagator matrix for a particle consists of its vacuum propagator and a term depending on the on-shell distribution function of like particles in the medium through which it propagates. The form of the latter term depends only on whether the particle in question is a boson or a fermion.

The 11-component of the thermal pion propagator is given by

 D11(k,mπ)=Δ(k,mπ)+2πiN21(k,mπ)δ(k2−m2π) (1)

where

 Δ(k,m) = −1k2−m2+iη N1(k,m) = θ(k0)√nk++θ(−k0)√nk− (2)

with

 nk±=1eβ(ωk∓μk)−1  ;  ωk=√→k2+m2

and for and 0 for .

The thermal part remaining same, the propagator with the familiar polarization sum follows as

 Dμν11(k,mρ)=(−gμν+kμkνm2ρ)D11(k,mρ) . (3)

We now consider the fermionic propagators whose thermal (matrix) parts are different from the bosonic ones discussed above. The 11-components of the nucleon and propagators are respectively given by

 S11(p)=(p+mN)E11(p,mN) (4)

and

 S11μν(p)=(p+mΔ) {−gμν+23m2Δpμpν+13γμγν (5) +13mΔ(γμpν−γνpμ)}E11(p,mΔ)

where, is given in Appendix-A.

The complete propagator is given by the Dyson equation in terms of the free fermion propagator and self-energy ,

 S′=S−SΠS′ (6)

where each is a 22 matrix in the thermal indices. They can be diagonalized to get the respective analytic functions, denoted by bar, so that

 ¯¯¯¯S′=¯¯¯¯S−¯¯¯¯S¯¯¯¯Π¯¯¯¯S′ . (7)

The self-energy function can be obtained from any single component of the self energy matrix as discussed in Appendix-A. It is related to, say, the 11-component by

 Im¯Π(p) = ϵ(p0)coth[β(p0−μp)/2]ImΠ11(p) Re¯Π(p) = ReΠ11(p) (8)

where for and for .

### ii.2 The Δ self-energy

We begin by writing down the expressions for the one-loop self-energy graphs for the in vacuum. For the four cases shown in fig. 1 they can be generally expressed as

 Πμνvac(q)=i∫d4k(2π)4NμνΔ(p)Δ(k) (9)

where contains terms coming from the two vertices and the spin factors appearing in the propagators for the internal lines. They can be read off from the expressions for given in Appendix-B and appear as

 NμνπNΔ = f2πNΔm2πF2(p,k)[kαkβOνβ(p+mp)Oαμ] (10) NμνρNΔ = (11) NμνπΔΔ = f2πΔΔm2πF2(p,k)[gχψgηϕOνχγ5kOψσΣλσ(p)Oληγ5kOϕμ] (12) NμνρΔΔ = f2ρΔΔF2(p,k)[gχψgηϕOνχ(γβ+iκρΔΔ2mΔσβϵkϵ)OψθΣλθ(p) (13) Oλη(γα−iκρΔΔ2mΔσαδkδ)OϕμAαβ]

where

 Aαβ(k) = −gαβ+kαkβm2k   and Σαβ(q) = (q+mq)[−gαβ+13m2qqαqβ+13γαγβ+13mq(γαqβ−γβqα)] . (14)

We now proceed to write down the corresponding expressions in the medium. As discussed above, we need to evaluate only the 11-component which is obtained by replacing the vacuum propagators by the 11-component of thermal propagators given in the previous section. The self-energy in the medium is thus given by

 Πμν11(q)=i∫d4k(2π)4NμνE11(p)D11(k) . (15)

Expanding and we obtain in addition to , two terms which are linear in the thermal distribution function and the fourth term non-linear in the distribution function which is purely imaginary. Performing the integral and using eqns. (8) we obtain the imaginary and real parts of the self-energy function as

 Im¯Πμν(q) = −πϵ(q0)∫d3k(2π)314ωkωp× [Nμν(k0=ωk){(1+nk+−~np+)δ(q0−ωk−ωp)+(−nk+−~np−)δ(q0−ωk+ωp)}+ Nμν(k0=−ωk){(−1−nk−+~np−)δ(q0+ωk+ωp)+(nk−+~np+)δ(q0+ωk−ωp)}]

and

 Re¯Πμν(q)=∫d3k(2π)312ωkωp P[(nk+ωpNμν(k0=ωk)(q0−ωk)2−ω2p)+(nk−ωpNμν(k0=−ωk)(q0+ωk)2−ω2p)− (~np+ωkNμν(k0=q0−ωp)(q0−ωp)2−ω2k)−(~np−ωkNμν(k0=q0+ωp)(q0+ωp)2−ω2k)]

where and .

Each of the terms in the imaginary part can be related to scattering and decay of the -baryon. The delta functions in the four terms define the kinematic domains where these processes occur. The regions where these are non-vanishing correspond to branch cuts in the complex plane. The first and third terms in (LABEL:im1) are non-vanishing for . This is the usual unitary cut already present in vacuum. The second and fourth terms which are non-zero for defines the Landau cut and is purely a medium effect. Confining ourselves to the kinematic region and , the first and fourth terms only contribute. The first corresponds to absorption of the due to decay into a baryon-meson pair such as , etc. and is thus weighted by a thermal factor , indicating Bose enhancement of the meson and Pauli blocking of the baryon in the process plus the usual thermal factors for the initial state in the time reversed process where the is produced. The fourth term is due to absorption of the due to scattering from a meson producing a baryon in the final state and vice versa as is evident from the thermal weight factor .

To take into account the finite width of unstable particles in the loop graphs the self-energy is folded with their spectral functions. As a consequence the sharp thresholds of the branch cuts get smeared. For unstable mesons () we use Gonzalez (),

 Π(q,mh)=1Nh∫(mh+2Γh)2(mh−2Γh)2dM21πIm[1M2−m2h+iMΓh(M)]Π(q,M) (18)

with and . Here so that

 Γρ(M)=Γρ→ππ(M)=[g2ρππ48πM3][M2−4m2π]λ12(M2,m2π,m2π) (19)

where . This is obtained using with .

For baryons () with non-trivial decay width in the loops we use

 Π(q,mR)=1NR∫mR+2ΓRmR−2ΓRdM1πIm⎡⎣1M−mR+i2ΓR(M)⎤⎦Π(q,M) (20)

with and . In this case for which the decay formula is given by eq. (22) in Section III.

### ii.3 Numerical results

Let us begin with the results for the self-energy in the medium. We show numerical results for the spin-averaged real and imaginary parts of the function given by Wehrberger (); Kim ()

 Π=14∑sΔ¯Ψμ¯ΠμνΨν (21)

where denote Rarita-Schwinger spinors. The factors given in eqs. (13) then go over to . In fig. 2 we plot the Landau cut contribution to the imaginary part coming from the different loop graphs at MeV for MeV and MeV in panels (a) and (b) respectively. The corresponding results for MeV are shown in panels (c) and (d). As discussed in the last section, it follows that for the Landau cut extends up to the difference of masses of the baryon and the meson in the loop graph in the stable limit. These contributions at lower values of are a result of scattering processes in the thermal medium and are absent in vacuum. Comparing fig. 2(a) with (b) (and (c) with (d)) we see that the and loops start contributing to the imaginary part only at larger baryon densities.

The unitary cut contributions to the imaginary part are much larger in magnitude than the ones coming from the Landau cut. The thresholds lie at higher energies, for stable particles starting from the sum of the nominal masses of the particles in the loop graph. Shown in fig. 3 are the unitary cut contributions from the four loops at MeV. We observe the sequential opening up of heavier decay channels. These are the same as in vacuum but are now weighted by the Pauli blocking and Bose enhancement factors in the final state. The form factor suppresses the usual monotonous rise of these contributions at higher .

We now consider the thermal component of the real part of the self-energy consisting of principal value integrals. As seen in fig. 4 the magnitudes are quite small compared to the imaginary parts and are not expected to contribute to a thermal shift in the pole position of the .

Having obtained the imaginary and real parts of the self-energy we now plot the spectral function which is the imaginary part of the complete propagator (7). We do so in two parts. The low region depicting the Landau cut contribution shown in panel (a) of fig. 5 is purely a thermal contribution. The high region in the vicinity of the bare mass consisting of the contributions from the unitary cuts is shown in panel (b) of fig. 5. The spectral density is seen to have a significant dependence on the temperature and chemical potential of the medium. However, comparing the curves at and MeV, both for MeV, the effect of baryon density only shows up at higher temperatures and has its origin in the thermal distribution functions for the baryons. In general, we find a gradual suppression of the peak with increasing temperature and density owing to the larger imaginary parts in the denominator of the in-medium propagator. As seen from eq. (LABEL:im1), the increase in the imaginary part comes from two factors: (i) the Bose enhancement factor for the pions and rho mesons in the final state in the first term which is the unitary cut contribution and (ii) the Landau cut contribution coming from scattering of the mesons off the propagating as given by the fourth term. The second and third terms do not contribute because of kinematic reasons.

In fig. 6(a) we have made a comparison of the spectral function obtained in our approach with that of vanHees () for two sets of values of temperature and nucleon density representative of conditions likely to be achieved in relativistic heavy ion collisions at RHIC and in the CBM experiment. A reasonable agreement is observed between the present work and that of vanHees () as depicted by the continuous lines and symbols respectively. In addition to differences in the Lagrangian and associated parameters used in the two approaches the disparities in the spectral functions in the two cases could arise due to contributions coming from higher order effects introduced through dressed nucleon and pion propagators in the self-energy considered in vanHees () wherein vertex corrections were included through Migdal parameters in the pion propagator.

To take into account the finite size of the vertices a phenomenological hadronic form factor has been introduced. The details are provided in Appendix-B. The numerical results presented here correspond to a monopole type form factor, denoted by Form Factor I in fig. 6(b) with MeV which produces a good fit to the phase shift data and cross section. We also plot the spectral function using Form Factor I with MeV and find a small reduction at the peak though the cross section remains largely unchanged. This is because in the spectral function the square of the form factor appears multiplicatively but in the cross-section its effect is largely canceled at the -channel pole position as seen from eq. (23) in the next section. For a comparison we also plot the spectral function using an exponential form factor denoted by Form Factor II with GeV (as used in Korpa1 () with GeV). No appreciable difference is found with the one with 600 MeV. All the plots in this figure correspond to MeV, MeV and MeV. The symbols denote the results of vanHees ().

At this point a few comments on the hadronic form factors at the vertices are in order. It is well known that in local field theory involving spin-3/2 fields the redundant degrees of freedom associated with unphysical spin-1/2 fields are eliminated through the Rarita-Schwinger constraints Rarita (). The interacting case is more complex and suffers due to participation of the spurious spin-1/2 components. A coupling consistent with gauge invariance of the Rarita-Schwinger field was constructed in Pascalutsa () preserving the correct number of degrees of freedom. However, unphysical behavior in the computed tree-level cross-section results if the reaction is cut off by standard hadronic form factors Vrancx (). In this work we have used the conventional vertex, which does suffer from a small presence of spin-1/2 components both in vacuum and in nuclear matter at saturation density Korpa1 (). As described above, the cut-off in the form factor used here was obtained by fitting the cross-section. The spectral functions evaluated using this (conventional) vertex were found Korpa3 () to differ slightly with the ones calculated using the consistent coupling discussed above if the same form factor is applied in both the cases. It was further shown that this difference could be eliminated if the additional momentum factor stemming from the higher derivative nature of the consistent interaction Pascalutsa () was compensated either by an additional form factor term or by adjusting the cut-off values of the original form factor.

## Iii The π-N Cross Section

Having studied the spectral modification of the in the medium we are now in a position to investigate how these changes affect the cross-section. We aim to set up a dynamical framework wherein medium effects can be implemented using thermal field theoretic methods and which at the same time is normalized to the experimental data in vacuum. We consider the interaction (46) and first check with the phase shift data Koch () defining with the partial wave amplitude given by . The decay width which follows from the imaginary part of eq. (42) is

 ΓΔ(E)=124π(fπNΔmπ)2F2(E)→p 3E2[(E+mN)2−m2π] (22)

where the c.m. momentum . As seen in fig. 7, a reasonable agreement is obtained using MeV and MeV.

Next we evaluate the matrix elements for elastic scattering in the isospin basis in which we replace the free vacuum propagator by an effective one containing the vacuum self energy due to the loop diagrams mentioned above. Averaging over isospin, the squared invariant amplitude for the process is given by

 ¯|M|2 = ∑(2I+1)|MI|2∑(2I+1) (23) = 13(fπNΔmπ)4⎡⎣F4(k,p)Ts∣∣s−m2Δ−Π∣∣2+F4(k,p′)Tu(u−m2Δ)2 +2F2(k,p)F2(k,p′)Tm(s−m2Δ−ReΠ)3(u−m2Δ)∣∣s−m2Δ−Π∣∣2⎤⎦

where , and are given by

 Ts = Tr[(p′+mN)Ds(p+mN)γ0D†sγ0] (24) Tu = Tr[(p′+mN)Du(p+mN)γ0D†uγ0] (25) Tm = Tr[(p′+mN)Ds(p+mN)γ0D†uγ0] (26)

in which

 Ds = kαk′βOβνΣμν(qs)Oμα (27) Du = k′αkβOβνΣμν(qu)Oμα (28)

where is defined in (14).

The cross-section given by turns out to be in good agreement with the isospin averaged total elastic cross-section given in Prakash () (obtained using phase shift and inelasticity data from Bareyre () and Koch ())up to about 1.5 GeV as seen from the solid curve in fig. 8. It is to be noted at this point that we have considered only exchange in the evaluation of elastic scattering amplitude with the aim of fixing the parameters (see Appendix B) and thus obtaining a baseline for estimating the effect of the modified propagator on the cross-section. For a more general treatment it is necessary to consider the exchange of as well as nearby resonances like the Roper(1440), etc. in the evaluation of the scattering amplitudes. In such a case, however, it could be quite challenging to obtain a satisfactory agreement with the experimental data especially in the region beyond the peak.

Having thus normalized the framework with the experimental data we now turn on the medium effects. We replace the vacuum self energy in the above expressions by the in-medium ones evaluated in the real-time formalism described above. A significant suppression of the peak with increasing temperature is obtained owing to the increase in the imaginary part due to reasons explained earlier. The small upward shift at higher baryon densities comes from the small positive contribution of the real part of the self-energy. As seen in fig. 4, there are substantial cancellations between the contributions from various loops depending essentially on the attractive or repulsive nature of the effective interactions considered.

## Iv The shear viscosity of a π−N gas

Transport coefficients can be obtained in (a) the kinetic theory approach using the transport equation and (b) the diagrammatic approach using Kubo formulae which relates them to retarded two-point functions. The latter was used in Sabya () to obtain the shear viscosity of a pion gas. However, for our present purpose which is to highlight the effect of the in-medium cross-section on the shear viscosity, the kinetic theory approach is more suited (see e.g Mitra1 (); Mitra2 ()). The transport equation describing the evolution of the phase space density of pions and nucleons in a hadronic gas mixture slightly away from local equilibrium is given by

 ∂fn∂t+→vn⋅→∇fn=C[fn]      (n=π,N) (29)

where is the particle velocity and is the collision integral. The distribution function for such a system assumes a form where the local equilibrium distribution function is given by , the plus and minus signs correspond to nucleons and pions respectively and is the deviation function. , and denote the local temperature, fluid velocity and chemical potentials. Assuming all except the particle to be in equilibrium the collision integral simplifies to Gavin ()

 C[fn]≃−(fn−f0n)τn=−δfnτn (30)

where is the relaxation time which characterizes the rate of change of the distribution function due to interaction with the species in the medium. For binary elastic collisions it is Prakash ()

 [τn(pn)]−1= ∑l=π,N[τnl(pn)]−1 (31)

with

 [τnl(pn)]−1=gl1+δnlcsh(ϵn/2)En∫dωldω′ndω′lWnl (32)

where , and the function if represents a fermion (boson). The dynamical input which goes into the determination of the distribution function appears in the interaction rate .

To extract the shear viscosity we turn to the energy-momentum tensor. For small gradients of the local fluid velocity the shear dissipative part is well-known to be Gavin ()

 Tijshear=−η(∂ui∂xj+∂uj∂xi−23→∇⋅→uδij) . (33)

Now, in terms of the distribution function the correction to the component of the stress-energy tensor is given by Itakura ()

 Tijdiss=∑n=π,Ngn∫d3pn(2π)3Enpinpjnδfn . (34)

From (29) and (30) we get to lowest order

 δfn = −τn[∂f0n∂t+→vn⋅→∇f0n] (35) = τnf0n2TEn(1±f0n)pinpjn(∂ui∂xj+∂uj∂xi−23→∇⋅→uδij)

where in the last line we retained only the (traceless) part appropriate for shear viscosity. Putting this in (34) and equating with (33) we obtain the shear viscosity of the pion-nucleon mixture,

 η=115T∑n=π,N∫d3pn(2π)3τn(pn)E2n|→pn|4f0n(1±f0n) . (36)

The shear viscosity is thus made up of contributions from the pion and nucleon components which are coupled through the momentum dependent relaxation times and . The relative importance at a given value of and is a consequence of interplay between the phase space factors as well as scattering cross-sections. In order to focus on the effect of the in-medium cross-section we take the and cross-sections in vacuum.

The mean relaxation time of the specie is defined in terms of the thermal average of the momentum dependent inverse relaxation time . With

 ¯¯¯ωi(T,μ)=∫d3k ωi(k)f0i(k)/∫d3kf0i(k) , (37)

the mean relaxation time is given by . We plot in fig. 9(a) the mean relaxation time of nucleons as a function of for two values of with and without medium effects. The features of the numerical results can be understood by realizing that the relaxation time for binary collision approximately goes as so that . Thus, the increase of density of the species, in this case pions, with plays the dominant role and accounts for the decreasing nature of the curves. It also follows that the relative increase of nucleon density for larger results in a relative decrease of the relaxation time. Again, since the pion density is considerably more than the nucleon density and is also much larger than , the nucleon relaxation time is expected to be dominated by the pion component. Since is smaller compared to vacuum, as seen in fig. 8, the relaxation time is larger in the medium. Similar arguments hold also for the pion relaxation time. Its magnitude is decided by the (vacuum) cross-section which being much larger, overshadows the medium dependence of the cross-section. We thus do not show it separately. These features as well as the results with the vacuum cross-section are quite in agreement with Prakash ().

In fig. 9(b) the shear viscosity is plotted as a function of . As discussed above the behavior of the pion and nucleon components and their relative magnitude decides that of the mixture. For lower nucleon densities the pion component dominates the viscosity of the mixture. As the nucleon density increases there is a substantial increase in the nucleon component and a decrease in the pion component, the sum being more for MeV compared to 200 MeV. This feature is irrespective of the cross-section (vacuum or medium) and is due to the interplay between the relative abundance of the species and the magnitude of their interaction cross-section. The fact that the shear viscosity in the medium is more than that in vacuum can be understood in terms of the relaxation times which are larger basically due to the lower cross-section in the medium. The increase in magnitude of the medium effect with temperature and nucleon density can be attributed to the phase space factors.

## V Summary and Outlook

In this work we have studied the spectral modification of the baryon in the medium. The self-energy was evaluated from one-loop graphs comprising of , , and using the real-time formulation of thermal field theory. In addition to the contributions from decay processes occurring above thresholds which arise due to the usual thermally weighted unitary cut in the complex plane there are significant contributions coming from the Landau type discontinuities in the low region stemming from scattering processes leading to the absorption of in the medium. The cross-section is then evaluated with the effective propagator of the leading to a suppression at finite temperature and density with no significant shift of the peak position. This is expected to have non-trivial consequences on the mean free path and should consequently affect the thermalization rate of pions Barz () and nucleons produced in heavy ion collisions. We finally make an estimate of the shear viscosity of a gas of pions and nucleons using the kinetic theory approach and observe an enhancement corresponding to the in-medium cross-section which increases with temperature and nucleon density. When used as input in the hydrodynamic equations this is expected to have an observable consequence of the space-time evolution of the latter stages of heavy ion collisions.

## Vi Appendix

### vi.1 Diagonalising the fermion propagator

The procedure of diagonalization of the thermal matrices appearing in the real time formulation of thermal field theory only concerns the bosonic or fermionic nature of the field Mallik (). The spin sum which appears in the numerator of the propagator can thus be factored out. The free thermal fermion propagator matrix is written as where the matrix has components

 E11=−E∗22=Δ(p,m)−2πi~N21δ(p2−m2) E12=−2πieβμ/2~N1~N2δ(p2−m2) E21=2πie−βμ/2~N1~N2δ(p2−m2) . (38)

The terms and containing thermal factors are given by

 ~N1(p0) =θ(p0)√~np++θ(−p0)√~np− ~N2(p0) =θ(p0)√1−~np+−θ(−p0)√1−~np− (39)

where

 ~np±=1eβ(ωp∓μp)+1 ,   ωp=√→p2+m2 .

The matrix can be diagonalized as

 E=V(Δ00−Δ∗)V (40)

where

 V=(~N2−~N1eβμ/2~N1e−βμ/2~N2) . (41)

It can be shown Mallik () that the complete thermal propagator matrix is also diagonalized by . From the Dyson equation (6) it then follows that the self energy matrix is diagonalized by resulting in eq. (7). The diagonal element is given by any one of the four components of as given in eq. (8).

### vi.2 The Δ self-energy in vacuum : Lagrangian and parameters

The self energy in vacuum for the one loop diagrams shown in fig. 1 are given by

 ΠμνπN = if2πNΔm2π∫d4k(2π)4F2(p,k)OνβkβS0(p)OαμkαD0(k) (42) ΠμνρN = if2ρNΔm2ρ∫d4k(2π)4F2(p,k)Oνηγ5γϕ(gβϕkη−gβηkϕ)S0(p)γ5γλ(gαλkσ−gασkλ)OμσDαβ0(k) (43) ΠμνπΔ = if2πΔΔm2π∫d4k(2π)4F2(p,k)Oνχγ5γβkβOψσgχψS0λσ(p)Oληγ5γαkαOϕμgηϕD0(k) (44) ΠμνρΔ = if2ρΔΔ∫d4k(2π)4F2(p,k)Oνχ(γβ+iκΔΔρ2mΔσβϵkϵ)Oψσgχψ (45) S0λσ(p)Oλη(γα−iκΔΔρ2mΔσαδkδ)OϕμgηϕD0αβ(k)

where and are the scalar and vector propagators in vacuum. The ones for the spin 1/2 and 3/2 fermions are given by and respectively. The vertex factors come from the well-known interactions Krehl ()

 LπNΔ = fπNΔmπ¯ΔαOαμ→T†∂μ→πψ+H.c. (46) LρNΔ = −ifρNΔmρ¯ΔαOαμγ5γν→T†→ρμνψ+H.c. (47) LπΔΔ = fπΔΔmπ¯ΔαOαμγ5γν→TΔμ∂ν→π (48) LρΔΔ = −fρΔΔ¯ΔβOαβ[γμ−κρΔΔ2mΔσμν∂ν]→ρμ→TΔα (49)

where Krehl (); Korpa1 (), , , , and . In the above

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• March 05, 2021, 01:27:04 PM

### Author Topic:  How to Find An Optimal Number of Maximum Iterations  (Read 575 times)

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#### Byte11

• Fractal Fanatic
• Posts: 39
##### How to Find An Optimal Number of Maximum Iterations
« on: December 22, 2017, 12:13:52 AM »
I'm looking to find the optimal number of maximum iterations in the most computationally cheap way. I had a method where I would take a point (usually the center) and store it in full precision. Then I would limit the precision of the reference calculation to $$precision = frame + 15$$ with precision in bits (this is assuming magnification is given by 2^frame). Then I'd calculate the reference until the point tended towards 0 or infinity. The number of iterations that took would be the optimal number of iterations. However, this only worked given two things: 1. The point being calculated was relatively complex compared to the rest of the pixels 2. I had more bits of precision in my reference than I was using for my zooming.

This worked well if you take a point from a previous zoom and plug that into the formula. However, if the user is clicking on pixels and I convert those pixels to a new center point to use as a reference for this algorithm, the point is almost always unsuitable.

So I either need a way of finding points that fit the above criteria or another algorithm for finding an optimal number of iterations. For finding a suitable point, I was thinking to either use Misiurewicz points or Newton-Raphsoon zooming, but those are computationally expensive. Does anyone else have a better idea?

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#### claude

• 3f
• Posts: 1792
##### Re: How to Find An Optimal Number of Maximum Iterations
« Reply #1 on: December 22, 2017, 01:25:13 AM »
in my mandelbrot-perturbator I do iterations in chunks: say 1000 reference iterations, then 1000 pixel iterations, repeat.  That way I mostly don't need to know the ideal maximum iteration count before starting, and can set it ludicrously high as it will stop when all pixels have escaped (the exception is when minibrots are visible, in which case some pixels would never escape - I will eventually extend mandelbrot-perturbator either to do interior checking or to keep doubling the iteration count until the rate of pixels escaping is low, a strategy I learned from Robert Munafo).

• Fractal Frogurt
• Posts: 452
##### Re: How to Find An Optimal Number of Maximum Iterations
« Reply #2 on: December 22, 2017, 04:02:03 PM »

#### Byte11

• Fractal Fanatic
• Posts: 39
##### Re: How to Find An Optimal Number of Maximum Iterations
« Reply #3 on: December 24, 2017, 01:26:47 AM »
The algorithm that I described above works well if the selected point is significantly deeper than the points in the viewing area but is still within the bounds of the viewing area. Is there a way to find points within the viewing area that are fairly deep that could be used for this algorithm?

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Home » Posts tagged 'distance'

# Tag Archives: distance

## Doubt in Law of Gravitation

According to law of gravitation
F=g*(m1*m2)/r^2
If we bring two objects close to each other say our hands then the distance will be zero and the force of attraction will become infinite because r will we zero. Therefore F will become infinite but this is not the case.
When we touch our both palms together we can easily pull then away.

When we consider the distance, it is between the center of mass of the two bodies under consideration. When we bring our hands together the distance between their CM is not zero.

Other wise, the two particles under consideration must be point masses.

Then if you consider two atoms or two nuclei, the forces that come into play will be different if we decrease the distance between two atoms or nuclei. The attraction will turn into strong repulsion when the limits are exceeded.

## Problem from Kinematics

The speed of a train changes from36km/h to 72km/h in 10sec. calculate the distance travelled during this time

u=36km/h = 10 m/s

v=72 km/h=20 m/s

t=10 s

S=?

use v = u+ at to find a

a = 1 m/s2

Use v2-u2=2aS to find S

S = (400-100)/2 = 150 m

## Ron writes on Light and Relativity

Let me start by saying I am NOT a physicist.  That much should be painfully clear soon enough.  Please bear with me as I am not certain I have the language and in depth knowledge to explain myself adequately.

A few years ago I began experiencing parallels between the world of human behavior and physics.  At the time it seemed simple enough.  That is until I started teaching myself physics.  But at its core it seemed to make sense, the rules of life recycle themselves in different forms.  Out of simple ideas comes complexity.  Newton’s laws, thermodynamics… pascals principle, snells law, red and blue shift, wave particle duality and on and on… all seem to have their parallels in human behavior and on the surface seem to follow the same mathematical equations. Then I ran into the most famous of physics equations E=MC^2.  It makes sense in human behavior.  The energy (work) we can get out of an individual is relative to that individuals’ mass.  The more matter we attach to the individual, knowledge, life experience etc (therefore increasing their mass) the more we can get out of them in the form of energy… but then the problem of the speed of light squared.  The only known constant (light speed) is a problem in human behavior.  At least to this point I know of no known behavioral constant.

I juggled the idea of it being a relative constant.  Constant for the individual but relative as it would differ for everyone.  The speed of cognition, or thought speed, would remain the same in potential throughout life of a given individual.  Or that it was just a theoretical potential that humanity had yet to obtain.  I even juggled the idea of it being a collective ability, but all of these options cause problems with the original equation.  This has forced me to contemplate the nature of light and left me with questions I simply lack the understanding of physics to answer.

Is the speed of light truly constant or is it only constant as it relates to the big picture?  As in: our perception of time as it relates to all time that has ever existed would appear as a single point in time.  The older we get the more time seems to “fly by”  if we as humans could continue to live for 13 billion years would the perception of an hour become so perceptively small that we wouldn’t even know it has passed?   Is it therefore possible we can only understand and therefore measure the speed of light at a specific point in time, even if we try to come back and remeasure and compare the speed of light now with the speed of light fifty years from now the difference between the two would be imperceptivity small as a result of the displacement in time as it relates to the whole of time?

If the four dimensionality of time/space is linked shouldn’t time expand as space expands? And vice versa… if the speed of light is to remain constant as measured under such conditions isn’t light actually slowing down/speeding up over time as it relates to the whole of existing time/space?  It’s a distance displacement problem…. If points A and B are actually farther apart but light travels the same distance in the same time the “speed” may seem constant but time has actually expanded to give the light more “time” to cover that distance.  The speed of light would therefore be constant as a relation to perception and not as it relates to physical principles. Like the fact that the perception of time changes as we get older even though the actual measure of time remains the same.

How can light exist forever?  At the speed of light we theoretically freeze time for that photon but that would also require an infinite amount of energy to obtain and maintain. So even a subatomic particle with a lifespan of a nanosecond would appear to exist for all eternity but in actuality would still only exist for a nanosecond.  Because quite simply…it can’t have infinite energy and if it can I don’t understand how.  Thermodynamics: no system is a perfect system and will experience energy loss, Newton: equal and opposite reaction, if something begins it has to end to balance the equation.

Is it therefore possible that light is born of the fourth dimension… we experience it in the dimensionality of space as long as it loses its energy to the three dimensions.  A photon folds and pushes its way through space/time  The initial energy of the photon is high and generates bigger leading waves which resist the photon holding it from passing the “speed of light” as the energy dissipates/photon begins to die those folds restricts the photon less allowing it to maintain the speed of light.  We experience light because of its “ripples” in three-dimensions light dissipates as the waves of space become less folded in front of the photon.  But this would mean the photon eventually loses enough energy that it can no longer be perceived in the third dimension….  So what happens to it??  Imagine : A man running through a corn field has to exert the energy to push the stalks aside but  over time if the corn stalks are slowly spaced out even as he loses energy he can maintain his pace, because there is less impeding his path.  Our perception of light would be like being in a helicopter looking down on the field.  In the begging the field is densely packed with cornstalks and they slowly spread out until there is none.  We only know the man is there while he is running through the corn because he pushes the stalks aside and we see that movement but once there are no stalks or he lacks the energy to continue running we no longer have a way of measuring his presence.

That brings me for some reason to a theory of Dark matter.  Why?  Well what happens to photons that no longer move with enough force to be visible.  Like that guy running from a cornfield into an empty field.  How would we know he is still there?  If it was an infinite number of guys all stopping in that empty field we would know they are there by the depression their weight leaves in the field.  Or more accurately by the stones in the field rolling towards a depression we cant actually see.  The thought: An infinite number of “massless” (or perceived mass less) subatomic particles would still have infinite mass.  Infinite mass would supply more then enough gravitational pull even spread out over infinite distance to cause the continued and speeding expansion of space as more and more visible photons “die” contributing its “dark masslessness”.  Any dark matter existing within the universe would act as force… a moving invisible mass existing only on the 4th dimension pushing upon any objects in its way.  But because energy propagated internally expanding outward in all dimensions would compound on the outside as those energy’s converge.

I hope you followed those questions.  I don’t know what these ideas would do to theoretical physics but it would allow for the relativity of cognitive speed between individuals and reopen the door to “what the hell is dark matter in relation to human behavior” but that’s something different all together.

## A Numerical for Practice

Akshay posted a numerical:
1. A disk of mass m=50g slides with zero initial velocity down an incline plane set at an angle 30 degrees to the horizontal. Having traversed the distance of 50 cm along the horizontal plane, the disc stops. Find the work performed by friction forces over the whole distance. Assume friction coefficient is 0.15 for both horizontal and incline planes.

## A Problem based on Friction

an unloaded bus can be stopped by applying brakes on straight road after covering a distance x. Suppose, the passenger add 50% of its weight as the load and the braking force remains uncharged, how far will the bus go after the application of brakes? (velocity of bus in both case is same)

## Problem from Kinematics

Devika posted:

There are two particles A and B moving with same velocity u such that initially there velocity are perpendicular to each other and distance between them is L.the velocity of A is always directed towards B.find the distance between them after a long time.

Ans will be posted soon

## Average Speed

“A car travels from A to B with a speed of 30km/h and returns back to A with a speed of 50km/h.Find distance,displacement,average speed and average velocity ifdistance between A and B is 100km.”

Distance 200 m

Displacement 0

Average velocity 0

Average speed =$\frac{2v_{1}v_{2}}{v_{1}+v_{2}}$ = 37.5 kmph

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